where \(u=(u^1, \dots , u^m): D \times [0,T] \rightarrow \mathbb {R}^m\)  and \(F = (F^{ij}): \mathbb {R}^m \rightarrow \mathbb {R}^{m\times n}\) .
The system of conservation laws (REF ) is called hyperbolic, provided that the Jacobian \(D_u(F \cdot n)\)  possesses real eigenvalues for all unit vectors \(n\in \mathbb {R}^n\)  with \(|n|=1\) .
A great variety of systems in continuum mechanics can be formulated as hyperbolic systems of conservation laws, include the compressible Euler equations of gas dynamics, the shallow water equations of oceanography, the Magneto-Hydro-Dynamics (MHD) equations of plasma physics, and the equations of nonlinear elastodynamics [1]}.
